On the law of homogeneous stable functionals
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 82-111.

Let $$ be the $$-functional of a stable Lévy process starting from one and killed when crossing zero. We observe that $$ can be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections between $$ and more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties of $$ related to infinite divisibility, self-decomposability, and the generalized Gamma convolution.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2018028
Classification : 60G51, 60G52
Mots-clés : Beta random variable, exponential functional, homogeneous functional, infinite divisibility, stable Lévy process, time-change
Letemplier, Julien 1 ; Simon, Thomas 1

1 
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Letemplier, Julien; Simon, Thomas. On the law of homogeneous stable functionals. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 82-111. doi : 10.1051/ps/2018028. http://www.numdam.org/articles/10.1051/ps/2018028/

[1] K. Akita and M. Maejima, On certain self-decomposable self-similar processes with independent increments. Stat. Probab. Lett. 59 (2002) 53–59. | DOI | MR | Zbl

[2] E.W. Barnes, The theory of the G-function. Quart. J. Pure Appl. Math. 31  (1899) 264–314. | JFM

[3] E.W. Barnes, The theory of the double Gamma function. Phil. Trans. Royal Soc. London 196 (1901) 265–387. | JFM

[4] C. Berg and J.L. López, Asymptotic behaviour of the Urbanik semigroup. J. Approx. Theory 195 (2015) 109–121. | DOI | MR | Zbl

[5] J. Bertoin, On the local rate of growth of Lévy processes with no positive jumps. Stoch. Proc. Appl. 55 (1995) 91–100. | DOI | MR | Zbl

[6] J. Bertoin, Lévy Processes. Cambridge University Press, Cambridge (1996). | MR | Zbl

[7] J. Bertoin, Cauchy’s principal value of local times of Lévy processes with no negative jumps via continuous branching processes. Electr. J. Probab. 2 (1997) 1–12. | MR | Zbl

[8] J. Bertoin and M. Yor, On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electr. Commun. Probab. 6 (2001) 95–106. | MR | Zbl

[9] J. Bertoin and M. Yor, On the entire moments of self-similar Markov processes and exponential functionals. Ann. Fac. Sci. Toulouse VI. Sér. Math. 11 (2002) 33–45. | DOI | Numdam | MR | Zbl

[10] N.H. Bingham, Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. verw. Gebiete 26 (1973) 273–296. | DOI | MR | Zbl

[11] L. Bondesson, On the infinite divisibility of the Half-Cauchy and other decreasing densities and probability functions on the non-negative line. Scand. Actuarial. J. 1987 (1987) 225–247. | DOI | MR | Zbl

[12] L. Bondesson, Generalized Gamma convolutions and related classes of distributions and densities. In Vol. 76 of Lect. Notes Stat.. Springer-Verlag, New York (1992). | MR | Zbl

[13] L. Bondesson, A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. J. Theor. Probab. 28 (2015) 1063–1081. | DOI | MR | Zbl

[14] P. Bosch and T. Simon, On the self-decomposability of the Fréchet distribution. Indag. Math. 24 (2013) 626–636. | DOI | MR | Zbl

[15] P. Bosch and T. Simon, On the infinite divisibility of inverse Beta distributions. Bernoulli 21 (2015) 2552–2568. | DOI | MR | Zbl

[16] P. Bosch and T. Simon, A proof of Bondesson’s conjecture on stable densities. Ark. Mat. 54 (2016) 31–38. | DOI | MR | Zbl

[17] J.-P. Bouchaud, A. Comtet, A. Georges and P. Le Doussal Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. 201 (1990) 285–341. | DOI | MR

[18] M.E. Caballero and L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 (2006) 967–983. | DOI | MR | Zbl

[19] P. Carmona, F. Petit and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential functionals and principal values related to Brownian motion, edited by M. Yor. Biblioteca of RMI (1997) 73–121. | MR | Zbl

[20] L. Chaumont and M. Yor, Exercises in Probability. Cambridge University Press, Cambridge (2003). | DOI | MR | Zbl

[21] D.A. Darling, The maximum of sums of stable random variables. Trans. Am. Math. Soc. 83 (1956) 164–169. | DOI | MR | Zbl

[22] A. Diédhiou. On the self-decomposability of the Half-Cauchy distribution. J. Math. Anal. Appl. 220 (1998) 42–64. | DOI | MR | Zbl

[23] R.A. Doney, On the maxima of random walks and stable processes and the arc-sine law. Bull. London Math. Soc. 19 (1987) 177–182. | DOI | MR | Zbl

[24] D. Dufresne, The distribution of a perpetuity, with applications to risk theory and pension. Scand. Actuarial. J. 1990 (1990) 39–79. | DOI | MR | Zbl

[25] E.B. Dynkin, in Vol. I of Markov processes. Springer Verlag, Berlin (1965). | MR | Zbl

[26] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions in Vols. I and II. McGraw-Hill, New York (1953). | MR | Zbl

[27] W. Feller, The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat. 22 (1951) 427–432. | DOI | MR | Zbl

[28] P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 (1995) 3–58. | DOI | MR | Zbl

[29] P. Graczyk and T. Jakubowski, On exit time of stable processes. Stoch. Proc. Appl. 122 (2012) 31–41. | DOI | MR | Zbl

[30] D. Hackmann and A. Kuznetsov, Asian options and meromorphic Lévy processes. Finance Stochast. 18 (2014) 825–844. | DOI | MR | Zbl

[31] F. Hirsch and M. Yor, On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator. Bernoulli 19 (2013) 1350–1377. | DOI | MR | Zbl

[32] M. Jeanblanc, J. Pitman and M. Yor, The Feynman-Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 62 (1997) 27–52. | MR | Zbl

[33] W. Jedidi, T. Simon and M. Wang, Density solutions to a class of integro-differential equations. J. Math. Anal. Appl. 458 (2018) 134–152. | DOI | MR | Zbl

[34] J. Kent, Some probabilistic properties of Bessel functions. Ann. Probab. 6 (1978) 760–770. | DOI | MR | Zbl

[35] A. Kuznetsov, On extrema of stable processes. Ann. Probab. 39 (2011) 1027–1060. | DOI | MR | Zbl

[36] A. Kuznetsov, On the density of the supremum of a stable process. Stoch. Proc. Appl. 123 (2013) 986–1003. | DOI | MR | Zbl

[37] A. Kuznetsov, A.E. Kyprianou, J.C. Pardo and A.R. Watson, The hitting time of zero for a stable process. Electr. J. Probab. 19 (2014) 1–26. | MR | Zbl

[38] A. Kuznetsov and J.-C. Pardo, Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123 (2013) 113–139. | DOI | MR | Zbl

[39] A.E. Kyprianou and J.-C. Pardo, Continuous state branching processes and self-similarity. J. Appl. Probab. 45 (2008) 1140–1160. | DOI | MR | Zbl

[40] A.E. Kyprianou and V. Rivero, Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electr. J.Probab. 13 (2008) 1672–1701. | MR | Zbl

[41] J. Lamperti, Semi-stable Markov processes. I. Z. Wahrsch. verw. Gebiete 22 (1972) 205–225. | DOI | MR | Zbl

[42] J.B. Lawrie and A.C. King, Exact solution to a class of functional difference equations with application to a moving contact line flow. Eur. J. Appl. Math. 5 (1994) 141–157. | DOI | MR | Zbl

[43] J. Letemplier and T. Simon, Unimodality of hitting times for stable processes. Séminaire de Probabilités XLVI (2014) 345–357. | MR | Zbl

[44] J. Letemplier and T. Simon, The area of a spectrally positive stable process stopped at zero. Probab. Math. Stat. 38 (2018) 27–37. | DOI | MR | Zbl

[45] A. Nikeghbali and M. Yor, The Barnes G-function and its relations with sums and products of generalized Gamma convolution variables. Electr. Commun. Probab. 14 (2009) 396–411. | MR | Zbl

[46] D. Ostrovsky. Theory of Barnes Beta distributions. Electr. Commun. Probab. 18 (2012) 59. | MR | Zbl

[47] A.G. Pakes, Characterization by invariance under length-biasing and random scaling. J. Statist. Plann. Inference 63 (1997) 285–310. | DOI | MR | Zbl

[48] J.-C. Pardo, On the rate of growth of Lévy processes with no positive jumps conditioned to stay positive. Electr. Commun. Probab. 13 (2008) 494–506. | MR | Zbl

[49] P. Patie and M. Savov, Extended factorizations of exponential functionals of Lévy processes. Electr. J. Probab. 17 (2012) 38. | MR | Zbl

[50] P. Patie and M. Savov, Exponential functionals of Lévy processes: generalized Weierstrass products and Wiener-Hopf factorization. C. R. Math. Acad. Sci. Paris 351 (2013) 393–396. | DOI | MR | Zbl

[51] P. Patie and M. Savov, Spectral expansions of non-self-adjoint generalized Laguerre semigroups. To appear in: Memoirs Am. Math. Soc. (2019). | MR | Zbl

[52] S.C. Port, Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20 (1967) 371–395. | DOI | MR | Zbl

[53] B.A. Rogozin, The distribution of the first hit for stable and asymptotically stable random walks on an interval. Theory Probab. Appl. 17 (1972) 332–338. | DOI | MR | Zbl

[54] K. Sato, Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge (1999). | MR | Zbl

[55] T. Simon, Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs. Expo. Math. 28 (2010) 290–298. | DOI | MR | Zbl

[56] T. Simon, Hitting densities for spectrally positive stable processes. Stochastics 83 (2011) 203–214. | DOI | MR | Zbl

[57] R. Webster, Log-convex solutions to the functional equation f(x + 1) = g(x)f(x) : Γ-type functions. J. Math. Anal. Appl. 209 (1997) 605–623. | DOI | MR | Zbl

[58] K. Yano, Y. Yano and M. Yor, On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes. Séminaire de Probabilités XLII. In Vol. 1979 of Lecture Notes in Mathematics. Springer, Berlin  (2009) 187–227. | DOI | MR | Zbl

[59] M. Yor, On certain exponential functionals of real-valued Brownian motion. J. Appl. Prob. 29 (1992) 202–208. | MR | Zbl

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